JUST THE MATHS SLIDES NUMBER 13.6 INTEGRATION APPLICATIONS 6 - PDF document

“JUST THE MATHS” SLIDES NUMBER 13.6 INTEGRATION APPLICATIONS 6 (First moments of an arc) by A.J.Hobson 13.6.1 Introduction 13.6.2 First moment of an arc about the y -axis 13.6.3 First moment of an arc about the x -axis 13.6.4 The centroid of an arc

UNIT 13.6 - INTEGRATION APPLICATIONS 6 FIRST MOMENTS OF AN ARC 13.6.1 INTRODUCTION Let C denote an arc (with length s ) in the xy -plane of cartesian co-ordinates, and let δs be the length of a small element of this arc. Then, the “first moment” of C about a fixed line, l , in the plane of C is given by lim C hδs, � δs → 0 where h is the perpendicular distance, from l , of the ele- ment with length δs . ✡ ✡ ✡ ✡ ✡ ◗◗◗◗◗◗◗◗ ✡ ✡ ✡ h ✡ ✡ l ◗ δs r ✡ r ✡ ✡ C 1

13.6.2 FIRST MOMENT OF AN ARC ABOUT THE Y-AXIS Consider an arc of the curve, with equation y = f ( x ) , joining two points, P and Q, at x = a and x = b , respec- tively. Q y ✻ δs δy r r P ✲ x O a δx b The arc may divided up into small elements of typical length, δs , by using neighbouring points along the arc, separated by typical distances of δx (parallel to the x - axis) and δy (parallel to the y -axis). The first moment of each element about the y -axis is xδs . 2

Hence, the total first moment of the arc about the y -axis is given by lim C xδs. � δs → 0 But, by Pythagoras’ Theorem, � 2 �  δy   � ( δx ) 2 + ( δy ) 2 = � � � 1 + δs ≃ δx. �   �  δx Thus, the first moment of the arc becomes � 2 �  δy   x = b � � lim x = a x � 1 + δx � �   �  δx δx → 0 � 2 �  d y � b   � � = a x � 1 + d x. �   �  d x Note: If the curve is given parametrically by x = x ( t ) , y = y ( t ) , then, d y d y d t d x = . d x d t 3

Hence, � 2 + � � 2 � � d y � � d x 2 � �  d y   � � d t d t � � 1 + = , �   � d x  d x d t provided d x d t is positive on the arc being considered. If d x d t is negative on the arc, then the above formula needs to be prefixed by a negative sign. Using integration by substitution, � � 2 2 � �  d y  d y . d x � b   � t 2   � � � � � 1 + d x = � 1 + d t d t, a x t 1 x � �     � �   d x d x where t = t 1 when x = a and t = t 2 when x = b . Thus, the first moment of the arc about the y -axis is given by � 2 2 �  d x  d y � t 2     � � + d t, t 1 x ± �     �   � d t d t according as d x d t is positive or negative. 4

13.6.3 FIRST MOMENT OF AN ARC ABOUT THE X-AXIS (a) For an arc whose equation is y = f ( x ) , contained between x = a and x = b , the first moment about the x -axis will be � 2 �  d y � b   � � � 1 + d x. a y �   �  d x Note: If the curve is given parametrically by x = x ( t ) , y = y ( t ) , the first moment of the arc about the x -axis is given by � 2 2 �  d x  d y � t 2     � � + d t, t 1 y ± �     �   � d t d t according as d x d t is positive or negative. 5

(b) For an arc whose equation is x = g ( y ) , contained between y = c and y = d , we may reverse the roles of x and y in section 13.6.2 so that the first moment about the x -axis is given by � 2 �  d x   � d � � � 1 + d y. c y �   �   � d y  y ✻ S d δy δs R c ✲ x O rr δx 6

Note: If the curve is given parametrically by x = x ( t ) , y = y ( t ) , then the first moment of the arc about the x -axis is given by � 2 2 �  d x  d y � t 2     � � + d t, t 1 y ± �     �   � d t d t according as d y d t is positive or negative and where t = t 1 when y = c and t = t 2 when y = d . EXAMPLES 1. Determine the first moments about the x -axis and the y -axis of the arc of the circle, with equation x 2 + y 2 = a 2 , lying in the first quadrant. Solution Using implicit differentiation 2 x + 2 y d y d x = 0 . 7

d y d x = − x Hence , y. y ✻ ✡ ✡ a ✡ ✡ ✡ ✲ x O The first moment about the y -axis is given by � � 1 + x 2 � x � a � a � x 2 + y 2 d x. � � y 2 d x = 0 x � � 0 y But √ x 2 + y 2 = a 2 and y = a 2 − x 2 . Hence, ax � a first moment = √ a 2 − x 2 d x 0 � a � ( a 2 − x 2 ) � 0 = a 2 . = − a By symmetry, the first moment about the x -axis will also be a 2 . 8

2. Determine the first moments about the x -axis and the y -axis of the first quadrant arc of the curve with para- metric equations x = a cos 3 θ, y = a sin 3 θ. Solution d x d θ = − 3 a cos 2 θ sin θ and d y d θ = 3 a sin 2 θ cos θ. y ✻ ✲ x O The first moment about the x -axis is given by � 0 � 9 a 2 cos 4 θ sin 2 θ + 9 a 2 sin 4 θ cos 2 θ d θ. 2 y − π Using cos 2 θ + sin 2 θ ≡ 1, this becomes � π a sin 3 θ. 3 a cos θ sin θ d θ 2 0 = 3 a 2 � π sin 4 θ cos θ d θ 2 0 π  sin 5 θ = 3 a 2   2 = 3 a 2 5 .     5  0 9

Similarly, the first moment about the y -axis is given by � 2 2 �  d x  d y     � � π � + d θ 2 x �     � 0   � d θ d θ � π a cos 3 θ. (3 a cos θ sin θ ) d θ = 2 0 = 3 a 2 � π cos 4 θ sin θ d θ 2 0 π  − cos 5 θ   2 = 3 a 2     5  0 = 3 a 2 5 . Note: This second result could be deduced, by symmetry, from the first. 10

13.6.4 THE CENTROID OF AN ARC Having calculated the first moments of an arc about both the x -axis and the y -axis it is possible to determine a point, ( x, y ), in the xy -plane with the property that (a) The first moment about the y -axis is given by sx , where s is the total length of the arc; and (b) The first moment about the x -axis is given by sy , where s is the total length of the arc. The point is called the “centroid” or the “geometric centre” of the arc. For an arc of the curve, with equation y = f ( x ), between x = a and x = b , its co-ordinates are given by � 2 d x � � � 2 d x � � d y � � d y � b � b � 1 + � 1 + a x � a y � d x d x x = and y = . � 2 d x � � � 2 d x � � d y � � d y � b � b � 1 + � 1 + � � a a d x d x 11

Notes: (i) The first moment of an arc about an axis through its centroid will, by definition, be zero. In particular, let the y -axis be parallel to the given axis. Let x be the perpendicular distance from an element, δs , to the y -axis. The first moment about the given axis will be C ( x − x ) δs = C δs = sx − sx = 0 . C xδs − x � � � (ii) The centroid effectively tries to concentrate the whole arc at a single point for the purposes of considering first moments. In practice, the centroid corresponds, for example, to the position of the centre of mass of a thin wire with uniform density. 12

EXAMPLES 1. Determine the cartesian co-ordinates of the centroid of the arc of the circle, with equation x 2 + y 2 = a 2 , lying in the first quadrant Solution y ✻ ✡ ✡ a ✡ ✡ ✡ ✲ x O From an earlier example in this unit, the first moments of the arc about the x -axis and the y -axis are both equal to a 2 . Also, the length of the arc is πa 2 . Hence, x = 2 a and y = 2 a π . π 13

2. Determine the cartesian co-ordinates of the centroid of the first quadrant arc of the curve with parametric equations x = a cos 3 θ, y = a sin 3 θ. Solution y ✻ ✲ x O From an earlier example in this unit, d x d θ = − 3 a cos 2 θ sin θ and d y d θ = 3 a sin 2 θ cos θ. The first moments of the arc about the x -axis and the y -axis are both equal to 3 a 2 5 . Also, the length of the arc is given by � 2 2 �  d x  d y � a     � � + d θ − �     π �   � d θ d θ 2 � π � 9 a 2 cos 4 θ sin 2 θ + 9 a 2 sin 4 θ cos 2 θ d θ. = 2 0 14

This simplifies to π  sin 2 θ   2 = 3 a � π 3 a cos θ sin θ d θ = 3 a 2 . 2     0 2  0 Thus, x = 2 a and y = 2 a 5 . 5 15